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Divergent Series and Differential Equations Michèle Loday-Richaud Divergent series and differential equations Michèle Loday-Richaud To cite this version: Michèle Loday-Richaud. Divergent series and differential equations. 2014. hal-01011050 HAL Id: hal-01011050 https://hal.archives-ouvertes.fr/hal-01011050 Preprint submitted on 26 Jun 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Mich`ele LODAY-RICHAUD DIVERGENT SERIES AND DIFFERENTIAL EQUATIONS M. Loday-Richaud LAREMA, Universit´ed’Angers, 2 boulevard Lavoisier 49 045 ANGERS cedex 01 France. E-mail : [email protected] E-mail : [email protected] 2000 Mathematics Subject Classification.— M1218X, M12147, M12031. Key words and phrases.— divergent series, summable series, summability, multi- summability, linear ordinary differential equation. DIVERGENT SERIES AND DIFFERENTIAL EQUATIONS Mich`ele LODAY-RICHAUD Abstract.— The aim of these notes is to develop the various known approaches to the summability of a class of series that contains all divergent series solutions of ordinary differential equations in the complex field. We split the study into two parts: the first and easiest one deals with the case when the divergence depends only on one parameter, the level k also said critical time, and is called k-summability; the second one provides generalizations to the case when the divergence depends on several (but finitely many) levels and is called multi-summability. We prove the coherence of the definitions and their equivalences and we provide some applications. A key role in most of these theories is played by Gevrey asymptotics. The notes begin with a presentation of these asymptotics and their main properties. To help readers that are not familiar with these concepts we provide a survey of sheaf theory and cohomology of sheaves. We also state the main properties of linear ordinary differential equations connected with the subject we are dealing with, including a sketch algorithm to compute levels and various formal invariants of linear differential equations as well as a chapter on irregularity and index theorems. A chapter is devoted to tangent-to-identity germs of diffeomorphisms in C, 0 as an application of the cohomological point of view of summability. v Pr´epublications Math´ematiques d’Angers Num´ero 375 — Janvier 2014 CONTENTS 1. Introduction................................................... 1 2. Asymptotic expansions in the complex domain................ 5 2.1. Generalities...................................... ................. 5 2.2. Poincar´easymptotics. ............ 6 2.3. Gevrey asymptotics.................................. 17 2.4. The Borel-Ritt Theorem............................... 26 2.5. The Cauchy-Heine Theorem. 31 3. Sheaves and Cechˇ cohomology with an insight into asymptotics 37 3.1. Presheaves and sheaves.............................. 37 3.2. Cechˇ cohomology. 54 4. Linear ordinary differential equations: basic facts and infinitesimal neighborhoods of irregular singularities. 63 4.1. Equation versus system............................... 63 4.2. The viewpoint of -modules...................................... 65 D 4.3. Classifications. 71 4.4. The Main Asymptotic Existence Theorem. 87 4.5. Infinitesimal neighborhoods of an irregular singular point. 90 5. Irregularity and Gevrey index theorems for linear differential operators................................................... 99 5.1. Introduction...................................... 99 5.2. Irregularity after Deligne-Malgrange and Gevrey index theorems . 102 5.3. Wild analytic continuation and index theorems. 109 2 CONTENTS 6. Four equivalent approaches to k-summability. 111 6.1. First approach: Ramis k-summability.............................112 6.2. Second approach: Ramis-Sibuya k-summability. 118 6.3. Third approach: Borel-Laplace summation. 124 6.4. Fourth approach: wild analytic continuation. 155 7. Tangent-to-identity diffeomorphisms and Birkhoff Normalisation Theorem................................................... 159 7.1. Introduction...................................... 159 7.2. Birkhoff-Kimura Sectorial Normalization. 162 7.3. Invariance equation of g...........................................167 7.4. 1-summability of the conjugacy series h . 169 8. Six equivalent approaches to multisummabilitye . 171 8.1. Introduction and the Ramis-Sibuya series. 171 8.2. First approach: asymptotic definition. 174 8.3. Second approach: Malgrange-Ramis definition. 183 8.4. Third approach: iterated Laplace integrals. 186 8.5. Fourth approach: Balser’s decomposition into sums. 195 8.6. Fifth approach: Ecalle’s´ acceleration. 199 8.7. Sixth approach: wild analytic continuation. 206 Bibliography................................................... 211 Index of notations................................................... 217 Index................................................... 219 CHAPTER 1 INTRODUCTION Divergent series may diverge in many various ways. When a divergent se- ries issues from a natural problem it must satisfy specific constraints restricting thus the range of possibilities. What we mean, here, by natural problem is a problem formulated in terms of a particular type of equations such as differ- ential equations, ordinary or partial, linear or non-linear, difference equations, q-difference equations and so on . Much has been done in the last decades towards the understanding of the divergence of natural series, their classification and how they can be related to analytic solutions of the natural problem. The question of “summing” divergent series dates back long ago. Famous are the works of Euler and later of Borel, Poincar´e, Birkhoff, Hardy and their school until the 1920’s. After a long period of inactivity, the question knew exploding developments in the 1970’s and 1980’s with the introduction by Y. Sibuya and B. Malgrange of the cohomological point of view followed by works of J.-P. Ramis, J. Ecalle´ and many others. In these lecture notes, we focus on the best known class of divergent se- ries, a class motivated by the study of solutions of ordinary linear differential equations with complex meromorphic coefficients at 0 (for short, differential equations) to which they all belong. It is well-known (Cauchy-Lipschitz The- orem) that series solutions of differential equations at an ordinary point are convergent defining so analytic solutions in a neighborhood of 0 in C. At a singular point one must distinguish between regular singular points where all formal solutions are convergent (cf. [Was76, Thm. 5.3] for instance) and irregular singular points where the formal solutions are divergent in general; several examples of divergent series are presented and commented throughout 2 CHAPTER 1. INTRODUCTION the text. The strong point with formal solutions is that they are “easily” computed; at least, there exist algorithms to compute them, whatever the or- der of the linear differential equation. Nonetheless, one wishes to find actual solutions near such singular points and to understand their behavior. The idea underlying a theory of summation is to build a tool that trans- forms formal solutions into unique well-defined actual solutions. Roughly speaking, it is natural to ask that the former ones be linked to the latter ones by an asymptotic condition; in other words, that the formal solutions be Taylor series of the actual solutions in a generalized sense. Only convergent series have an asymptotic function on a full neighborhood of 0 in C; other- wise, the asymptotics are required on sectors with vertex 0. Uniqueness is essential to go back and forth and to guaranty good, well-defined properties. The problem is now fully solved for the class under consideration in several equivalent ways providing thus several equivalent theories of summation or theories of summability. Some methods provide necessary and sufficient con- ditions for a series to be summable, some others provide explicit formulæ. Each method has its own interest; none is the best and their variety must be thought as an enrichment of our means to solve problems. The theories here considered depend on parameters called levels or critical times. The simplest case with only one level k is called k-summability (actually, “simpler than the simplest” is the case when k = 1). The case of several levels k1,k2,...,kν is called multisummability or, to be precise, (k1,k2,...,kν)-summability. At first sight, since the singular points of differential equations are isolated, one could discuss the interest of such a procedure, for, one can approach as close as wished the singular points with the Cauchy-Lipschitz Theorem at the neighboring ordinary points. However, such an approach does not allow a good understanding of the singularities; even numerically, the usual numerical procedures stop being efficient when approaching a singular point, not providing thus even an idea of the behavior at the singular point. On the contrary, a good understanding of the singularity by means of a theory of summation permits a numerical calculation of solutions and of their invariants in most cases. Chapter 2 deals with asymptotics in the complex
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